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1-4 & 1-5 Angles Measures and Relationships
Objectives:
The student will be able to:
1. Measure and classify angles.2. Use congruent angles and the bisector of an
angle.3. Identify and use special pairs of angles.4. Identify perpendicular lines.
Classifying Angles
Acute Angles < 90° Right Angles = 90° Obtuse Angles > 90°
When naming angles using 3 letters, the vertex must be the second of the 3 letters. You can name an angle using a single letter only when there is exactly one angle located at the vertex.
Naming Angles:
Naming angles.
3
In the figure, QS is the angle bisector of . Point S lies in the interior of and . If and , find the value of x.
A ray that divides an angle into two congruent angles. PQS ≅ TQS The bisector of PQT is QS .
Congruent Angles & Angle Bisector:
50 = 4x + 14-14 -14
36 = 4x9 = x
In the figure, QS is the angle bisector of . Point S lies in the interior of and . If and , find the value of .
Example:
6x - 2 = 3x + 13+ 2
3x = 15x = 5
-3x
Did we answer the question? NO!
PQTTQSPQS
6(5) – 2 + 3(5) + 13 = PQT
30 – 2 + 15 + 13 = PQT 56° = PQT
If and , find the value of .
Special Angle pairs
Adjacent Angles:
Vertical Angles:
Linear Pair: 1 4, 2 3, 5 8, 6 7
Two angles that are opposite angles. Vertical angles are congruent.
1 & 2 , 2 & 4 , 4 &3, 3 & 1,
5 & 6, 6 & 8, 8 & 7, 7 & 5
Supplementary angles that form a line (sum = 180)
1 23 4
5 6
7 8
Two angles that lie in the same plane and have a common vertex and a common side, but no common interior points. 1 & 2, 1 & 3, 2 & 4, 3 & 4, 5 & 6, 5 & 7, 6 & 8, 7 & 8
Special Angle pairs
Congruent Angles:
Perpendicular angles:
Two or more angles that have the same measure.
Lines, segments, and rays that form right angles (90 degrees).
AEB & BEC, CED & DEA, AEB & DEC, BEC & AED
E
AEC & BED
Complementary & Supplementary Angles
Complementary Angles:
Supplementary Angles:
Two angles whose measures have a sum of 90°.
Two angles whose measures have a sum of 180°.
A + B = 30 + 60 = 90
F + G = 120 + 60 = 180
Identify:
Two Obtuse vertical angles:
Two acute adjacent angles:
An angle supplementary to TNU:
Find x so that .
If the two angles are perpendicular they MUST = 90° .
(9x + 5) + (3x + 1) = 90
12x + 6 = 90- 6 -6
12x = 84 x = 7
= 18021
x + (x – 18)= 180
2x – 18 = 180+18 +18
2x = 198x = 99
Example:
Find the measures of 2 supplementary angles if the difference in their measures is 18.
Are we through? NO!!
If x = 99, what are the measures of the supplementary angles?
99 99 -18 = 81
99 + 81 = 180
How can I check to see if that’s correct?
Find x and y so that KO and HM are perpendicular.
(3x + 6) + (9x) = 90
12x + 6 = 90- 6 -6
12x = 84x = 7
1. Find x by setting the two angles equal to 90.
90
2. Vertical angels tell us if , then .3. Find y by setting .
(3y + 6) = 90- 6 -6
3y = 84y = 28
1. Are the angles congruent? Yes – set the expressions equal to each other.
A = B
2. Do the angles add up to 90°?Yes – add the expressions and set them equal to 90°.
A + B = 90
3. Do the angles add up to 180°?Yes – add the expressions and set them equal to 180°.
A + B = 180
4. Do the angles add up to some other value given in the problem?
Yes – add the expressions and set them equal to the value.
A + B = other value
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